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64 CURVILINEAR COORDINATE SYSTEMS
(c) The field lines for v can be described by a function p = ~ ( 4 ). Show that
p ( 4 ) satisfies the equation
d P ( 4 ) - PV,
d 4 v4'
(d) Solve the above equation for ~ ( 4 ). and plot the v field lines on the same
- A flat disk rotates about an axis normal to its plane and passing through its center.
graph you plotted the surfaces of constant @.
Show that the velocity vector V of any point on the disk satisfies the equations
vTi=o V X T i = 2 W ,*
where W is the angular velocity vector of the disk. This vector is defined as
- Consider a sphere of radius r,, rotating at a constant angular rate w, about its z-axis so that
(a) Find an expression for the velocity vector V of points on the surface of the
sphere using spherical coordinates and spherical basis vectors. Remember v = w x i.
(b) Perform the integration
around the equator of the sphere. (c) Now perform the surface integral
j d W X V
over the entire surface of the sphere.
- The magnetic field inside an infinitely long solenoid is uniform
B = BOG,.
Determine the vector potential h such that v X h = B. For this situation, the magnetic field can also be derived from a scalar potential, B = --Fa. Why is this the case and what is @?
EXERCISES 65
The magnetic field inside a straight wire, aligned with the z-axis and carrying a uniformly distributed current, can be expressed using cylindrical coordinates as
where B, and po are constants. In this case the magnetic field can still be obtained from a vector potential v X A = g. Find this A. Now, however, it is no longer possible to find a scalar potential such that
21. A static magnetic field is related to the current density by one of Maxwell’s equations,
= -v@. Why is this the case?
v x B = poJ.
(a) If the magnetic field for p < po is given, in cylindrical coordinates, by
how is the current density 1 distributed?
shell at p = po?
(b) If B = 0 for p > p,, how much current must be flowing in a cylindrical
22. Classically, the angular momentum is given by
L = F X P ,
where F is the position vector of an object and
letting a vector differential operator Top act on a wave function q :
is its linear momentum. In quantum mechanics, the value of the angular momentum is obtained by
Angular Momentum = Top9.
The angular momentum operator can be obtained using the correspondence prin- ciple of quantum mechanics. This principle says that the operators are obtained by replacing classical objects in the classical equations with operators. Therefore,
Rep = F.
The position operator is just the position vector
The momentum operator is
pop = -iiiV,
where i is the square root of - 1 and fi is Planck’s constant divided by 27-r. Show that, in Cartesian geometry,
